Explanation

(c)
$I = \int_{}^{}\frac{dx}{x(\log{x)(\log{\log{x)\ldots\underset{8\ times}{\overset{(\log{\log{\ldots x)}}}{︸}}}}}}$
Put $\underset{8\ times}{\overset{(\log{\log{\ldots x)}}}{︸}} = t$
$\Rightarrow I = \frac{1}{\underset{7\ times}{\overset{(\log{\log{\ldots x)}}}{︸}}\ldots(\log{\log{x)(\log{x)x}}}}dx = dt$
$\therefore I = \int_{}^{}{\frac{dt}{t} = \log{t + c = \underset{9\ times}{\overset{(\log{\log{\log\ldots}}}{︸}}x) + c}}$